Hyperbolic curvature of holomorphic level curves
Mihai Iancu, Veronica-Oana Nechita
TL;DR
The paper addresses how the hyperbolic curvature of level curves associated with holomorphic self-maps $f:D\to D$ behaves, particularly for the level sets $\Omega(f)$ and $\Omega_\lambda(f)$. It develops sharp lower bounds for the hyperbolic curvature $k_h$ of their boundaries via Schwarz–Pontryagin–type arguments, reveals a rigidity phenomenon: if $k_h$ vanishes at a boundary point, then $f$ must be an automorphism and the boundary is a hyperbolic geodesic, and determines the radius of Euclidean convexity for the Jordan sublevel sets $\Omega(rf)$ as $\omega=1/\sqrt{2}$. The results also yield sharp estimates for the total hyperbolic curvature, hyperbolic area, and hyperbolic perimeter of sublevel sets, including a sharp hyperbolic isoperimetric inequality. Collectively, these findings provide precise geometric control over the hyperbolic geometry of level and sublevel sets under holomorphic maps of the disk, with exact equality cases identifying automorphisms.
Abstract
We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence, we point out a rigidity theorem: if the hyperbolic curvature of the above level curve vanishes at some point, then the level curve is a hyperbolic geodesic and $f$ is an automorphism. As another consequence, we prove that $\frac{1}{\sqrt 2}$ is the greatest lower bound of the supremum $r\in(0,1)$ such that the level curve $|z|=r|f(z)|$ is (Euclidean) convex. This constant turns out to be also the radius of convexity for hyperbolically convex self-maps of $\mathbb{D}$ that fix the origin. We also give (sharp) estimates for the total hyperbolic curvature, hyperbolic area and hyperbolic perimeter of the sublevel sets.